# The Dunkl-Coulomb problem in the plane

@article{Genest2014TheDP, title={The Dunkl-Coulomb problem in the plane}, author={Vincent X. Genest and A. Lapointe and L. Vinet}, journal={arXiv: Mathematical Physics}, year={2014} }

The Dunkl-Coulomb system in the plane is considered. The model is defined in terms of the Dunkl Laplacian, which involves reflection operators, with a $r^{-1}$ potential. The system is shown to be maximally superintegrable and exactly solvable. The spectrum of the Hamiltonian is derived algebraically using a realization of $\mathfrak{so}(2,1)$ in terms of Dunkl operators. The symmetry operators generalizing the Runge-Lenz vector are constructed. On eigenspaces of fixed energy, the invariance… Expand

#### 18 Citations

Algebraic approach to the Dunkl–Coulomb problem and Dunkl oscillator in arbitrary dimensions

- 2021

The Dunkl–Coulomb and the Dunkl oscillator models in arbitrary space-dimensions are introduced. These models are shown to be maximally superintegrable and exactly solvable. The energy spectrum and… Expand

Exact Solution of the Relativistic Dunkl Oscillator in $(2+1)$ Dimensions

- Physics, Mathematics
- 2018

In this paper we study the $(2+1)$-dimensional Dirac-Dunkl oscillator coupled to an external magnetic field. Our Hamiltonian is obtained from the standard Dirac oscillator coupled to an external… Expand

Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

- Mathematics, Physics
- 2016

We generalize the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach. This is presented in the context of 1D systems admitting ladder… Expand

COULOMB SYSTEMS WITH CALOGERO INTERACTION

- 2016

Apart from the N Liouville integrals, it possesses also N− 1 additional constants of motion [2]. As a result, the system is maximally superintegrable. It possesses various integrable generalizations… Expand

Algebra of Dunkl Laplace-Runge-Lenz vector

- Mathematics, Physics
- 2019

We consider Dunkl version of Laplace-Runge-Lenz vector associated with a finite Coxeter group $W$ acting geometrically in $\mathbb R^N$ with multiplicity function $g$. This vector generalizes the… Expand

Non-Hermitian inverted Harmonic Oscillator-Type Hamiltonians Generated from Supersymmetry with Reflections

- Physics, Mathematics
- 2017

By modifying and generalizing known supersymmetric models we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians… Expand

The Dunkl–Coulomb problem in three-dimensions: energy spectrum, wave functions and h-spherical harmonics

- Physics
- 2019

The Dunkl–Coulomb system in three-dimensions is introduced. The energy spectrum and the wave functions of the system are solved by means of spectrum generating algebra techniques based on the Lie… Expand

Spherical Calogero model with oscillator/Coulomb potential: quantum case

- Physics, Mathematics
- 2016

We consider the quantum mechanics of Calogero models in an oscillator or Coulomb potential on the N-dimensional sphere. Their Hamiltonians are obtained by an appropriate Dunkl deformation of the… Expand

Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems

- Physics, Mathematics
- 2016

We propose the integrable N-dimensional Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems and construct their constants of motion via the Dunkl operators. Their Schr\"odinger equations… Expand

The Wigner-Dunkl-Newton mechanics with time-reversal symmetry

- Physics
- 2020

In this paper, we use the Dunkl derivative concerning to time to construct the Wigner-Dunkl-Newton mechanics with time-reversal symmetry. We define the Wigner-Dunkl-Newton velocity and… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

- Mathematics, Physics
- 2013

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry… Expand

The Dunkl oscillator in three dimensions

- Mathematics, Physics
- 2014

The isotropic Dunkl oscillator model in three-dimensional Euclidean space is considered. The system is shown to be maximally superintegrable and its symmetries are obtained by the Schwinger… Expand

The singular and the 2:1 anisotropic Dunkl oscillators in the plane

- Physics, Mathematics
- 2013

Two Dunkl oscillator models are considered: one singular and the other with a 2:1 frequency ratio. These models are defined by Hamiltonians which include the reflection operators in the two variables… Expand

The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients

- Mathematics, Physics
- 2013

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is… Expand

Spectrum (super-) symmetries of particles in a Coulomb potential☆

- Physics
- 1985

The Schrodinger equation for a spin-0 particle in the field of a dyon is obtained by dimensional reduction of the four-dimensional harmonic oscillator; the reduction is effected by imposing an… Expand

The Bannai–Ito algebra and a superintegrable system with reflections on the two-sphere

- Physics, Mathematics
- 2014

A quantum superintegrable model with reflections on the two-sphere is introduced. Its two algebraically independent constants of motion generate a central extension of the Bannai–Ito algebra. The… Expand

Superintegrability in Two Dimensions and the Racah–Wilson Algebra

- Physics, Mathematics
- 2014

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere is cast in the framework of the Racah problem for the… Expand

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

- Mathematics
- 1996

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for… Expand

Dunkl operators and a family of realizations of osp(1|2)

- Mathematics
- 2009

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3… Expand

Laguerre semigroup and Dunkl operators

- Mathematics, Physics
- Compositio Mathematica
- 2012

Abstract We construct a two-parameter family of actions ωk,a of the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Here k is a multiplicity function for the Dunkl operators,… Expand